Fused graphical lasso and fused multiple graphical lasso

Fused graphical lasso (FGL) [43] and fused multiple graphical lasso (FMGL) [46] maximize the following penalized log-likelihood function,

where S(k)=(Y(k)-1nkY¯(k))T(Y(k)-1nkY¯(k))/nk is the sample covariance matrix and P(Θ) is the penalty term with Θ = {Θ⁽¹⁾, …, Θ^(K)}. As mentioned earlier, the only difference between FGL and FMGL is in the penalty term, P(Θ). FGL considers a pairwise fused lasso penalty and FMGL considers a sequential fused lasso penalty which have the following forms,

where λ₁, λ₂ are non-negative tuning parameters. The first term of both P^FGL(Θ) and P^FMGL(Θ) is the lasso penalty used in the GL model [18] that controls the overall sparsity. The second term of both the penalties controls the similarity of the off-diagonal elements of the precision matrices between conditions. Note that the second term of P^FMGL(Θ) is different from that of P^FGL(Θ) since it only focuses on differences between two consecutive conditions. If there are only two conditions i.e., K = 2, P^FGL(Θ) = P^FMGL(Θ). For K = 3, writing the penalties as functions of λ₁, λ₂, we show that P^FGL(Θ, λ₁, λ₂) ≤ P^FMGL(Θ, λ₁, 2λ₂). For K > 3, we are able to establish a crude connection: PFGL(Θ,λ1,λ2)≤PFMGL(Θ,λ1,⌊K24⌋λ2) (S1 Text). P^FGL(Θ) encourages the same level of similarity between all the pairs of conditions and P^FMGL(Θ) encourages the same level of similarity between each consecutive pair of conditions. However, these assumptions may be violated in practical scenarios. For example, two different subtypes of tumor tissues can be more similar to each other than to a healthy tissue. Therefore, ideally the penalty term should be such that it penalizes the difference between the tumor subtypes more than it penalizes the difference between one of the tumor subtypes and the healthy tissue. Lyu et. al. [48] addressed this issue by incorporating a special weight term into P^FGL(Θ) which is discussed in the next section.